Lets assume I want to start to sync from the last 10.000 blocks.
What does the wallet need to refresh from the monero daemon or remote node? Something like block headers?
How big is that that needs to be transferred from daemon to wallet?
The wallet will get block and pruned tx data (that is, roughly everything except the signatures and range proofs). For current transactions, that's around less than 5% of the full block size I believe. So if you have 100 kB average block size, you'd get about less than about 50 MB for 10000 blocks. I think it's less than that, but let's say 5% to be conservative.
For reference, 10000 blocks is about 2 weeks' worth of blocks.